semilocally

Semilocally is a term used in topology to describe properties that hold in a neighborhood of each point of a space. The most prominent instance is semilocally simply connected, a condition that interacts with the theory of covering spaces and fundamental groups.

A topological space X is semilocally simply connected if for every point x in X there exists a neighborhood U of x such that the image of the fundamental group under the inclusion-induced homomorphism i*: pi1(U, x) -> pi1(X, x) is trivial. Equivalently, every loop lying entirely within U is null-homotopic in X. This local or semi-local condition is weaker than being locally simply connected, in which every point has a neighborhood that is simply connected within itself.

The relevance of semilocal simple connectedness lies in covering space theory. If a space X is connected, locally path-connected, and semilocally simply connected, then X possesses a universal covering space. Without the semilocal condition, a universal cover may fail to exist or have pathological properties. The Hawaiian earring is a standard example of a space that fails to be semilocally simply connected at a base point, illustrating that the condition can be strictly necessary for certain covering-space results.

Related notions include local simple connectedness (stronger) and local path-connectedness (often assumed alongside semilocal conditions). These properties influence the behavior of fundamental groups, covering maps, and the structure of universal covers.

In summary, semilocally refers to properties that hold in neighborhoods of points, with semilocally simply connected spaces playing a central role in classical covering space theory and the study of fundamental groups.